Statistics and Algorithms / GARCH Models / GARCH Models (Univariate) /

UV GARCH Fat-Tailed Distributions

In many cases the assumption of conditional normality cannot be maintained. RATS offers two choices as alternatives: the Student-t and the Generalized Error Distribution (GED). You choose which to use with the DISTRIB option, which has choices NORMAL (the default), T and GED. Both the Student-t and GED have a shape parameter, which determines their kurtosis, and a scale parameter, which determines the variance given the shape parameter. Since ARCH or GARCH models a variance, the t or GED density selects the scale to give that variance. For instance, the GED has density function with kernel (the density function without the integrating constants)

\begin{equation} \exp \left( {{ - {{\left( {|x|/b} \right)}^{2/c}}} \mathbin{/} 2 } \right) \label{eq:garch_geddensity} \end{equation}

where \(c\) is the shape and \(b\) the scale. The variance, given \(b\) and \(c\), is

\begin{equation} \frac{{2^c b^2 \Gamma (3c/2)}}{{\Gamma \left( {c/2} \right)}} \label{eq:garch_gedvariance} \end{equation}

Given a variance from the model and the value of the shape parameter, \eqref{eq:garch_gedvariance} is solved for \(b\), and (the log of) \eqref{eq:garch_geddensity} (including the integrating constants) is used in maximum likelihood estimation.

Both densities have the normal as a special case. For the GED, that’s with shape parameter \(c=1\); for the t, it’s with infinite degrees of freedom. The GED family includes both fat-tailed densities (\(c>1\)) and thin-tailed ones (\(c<1\)), though it's very rare to see a thin-tailed estimate with typical data. The t with finite degrees of freedom is always fatter-tailed than the normal. Note that, because the t has no variance if the degrees of freedom are less than or equal to 2, the density function (as rescaled by RATS) won’t be defined for those values.

You can either set a value for the shape parameter using the option SHAPE, or have it estimated, which is the default if you don’t include SHAPE. In GARCHUV.RPF, the following two instructions estimate a GARCH(1,1) with a t distribution for the errors. In the first, the degrees of freedom are pegged at 5, while in the second, they’re estimated.

garch(p=1,q=1,distrib=t,shape=5) / dlogdm

garch(p=1,q=1,distrib=t) / dlogdm

This can be combined with any of the other models and options. Note that while Nelson(1991) used the GED with his proposed EGARCH model, there is no reason you have to use it, and, in fact, most empirical work done with EGARCH uses the Normal and t distributions used with other model types.

If you estimate a very low degrees of freedom (shape) with the t (anything less than 4 is rather small, since 2 is the lower bound), there's a good chance that you have some problem with the data. (Likely one or two really major outliers or a largely flat series with discrete changes). See the discussion under "Data Preparation".

Output

This adds the Shape (with either "t degrees" or "GED shape") to the output. The first is for a user-input value (hence the 0 standard errors), while the second is for the estimated value. If you input the SHAPE, it will not be included in the %BETA vector and (for obvious reasons) you can't do any hypothesis tests with it. If you allow the shape to be estimated, it will be included in %BETA and related statistics like %STDERRS and %TSTATS.

GARCH Model - Estimation by BFGS

Convergence in 18 Iterations. Final criterion was 0.0000000 <= 0.0000100

Dependent Variable DLOGDM

Usable Observations 1866

Log Likelihood -2053.1056

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Mean(DLOGDM) -0.032486965 0.015670240 -2.07316 0.03815709

2. C 0.016119967 0.006097303 2.64379 0.00819844

3. A 0.115537458 0.021201214 5.44957 0.00000005

4. B 0.877880508 0.021560686 40.71672 0.00000000

5. Shape(t degrees) 5.000000000 0.000000000 0.00000 0.00000000

GARCH Model - Estimation by BFGS

Convergence in 23 Iterations. Final criterion was 0.0000000 <= 0.0000100

Dependent Variable DLOGDM

Usable Observations 1866

Log Likelihood -2046.8940

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Mean(DLOGDM) -0.028988887 0.015392760 -1.88328 0.05966235

2. C 0.015059570 0.005220164 2.88488 0.00391558

3. A 0.104416871 0.017524463 5.95835 0.00000000

4. B 0.875756941 0.020172946 43.41245 0.00000000

5. Shape(t degrees) 8.775517067 1.688970553 5.19578 0.00000020